proof. The conclusion (i) is proved in [1]} by using variation of parameter. Note that the origin \(O\)  is an linear isochronous center as \(\lambda =0\)  and the period function \(T(\xi ,0)\equiv \,2\pi .\) 
By the continuous
dependence on the initial value and parameter associated with the
solution of differential equation, the period of closed orbit
\(\Gamma _{\xi ,\lambda }\)  is the variation of \(2\pi \) , denoted by
\(T(\xi ,\lambda )=2\pi +\rho (\xi ,\lambda )\) . It is clear \(\rho (\xi ,0)=0\)  and \(\frac{\partial {\rho (\xi ,0)}}{\partial {\lambda }}=0\)  is equivalent to vanishing of the following integral
\(\int _0^{2\pi }[xg(x,y)-yf(x,y)]_{x=\xi \cos \theta ,y=\xi \sin \theta }\,d\theta =0.\) 
